The local ray-tracing method propagates a wavefront locally
through triangular cells of linearly varying velocities by following local rays.
Because the whole process of wavefront propagation involves tracing many
local rays, it is necessary to have an efficient ray-tracing algorithm.
Fortunately, the analytical solutions to the ray equations for
a linearly varying velocity field are well known, and the analytical
solutions of *R* and *J* for such a velocity field are also known.
Here I merely quote the results. The details of the derivations
can be found in Telford et. al. (1976) and Cervený (1981abc).

Suppose we know the attributes of
a wavefront, ,
at an initial point (*x _{0}*,

(4) |

(5) |

(6) |

(7) |

(8) |

Cervený (1981a) shows that in a velocity field of zero second order spatial derivatives, the curvature radius of a wavefront at a point on a ray is linearly proportional to the integration of the velocity field along the ray. Because a linearly varying velocity field has zero second-order spatial derivatives, the integration of the velocity along the circular ray path gives

(9) |

(10) |

vpraytrc
Ray-tracing in a linearly varying velocity field. The dash arrow shows
the direction of the velocity gradient. The solid arrows show the directions
of traveltime gradients that are tangential to the ray. Ray angles are
measured between the direction of the velocity gradient and the directions
of the ray.
Figure 2 |

When a ray reaches the boundary of two adjacent triangular cells,
the phase matching method can be used to trace the ray across
the boundary (Cervený, 1981c).
Traveltime and take-off angle are always continuous
across the boundary. Because of the continuous
velocity representation, there is no discontinuity of velocity across
the boundary. Therefore traveltime gradient (*u*,*w*) and geometrical spreading factor
*J* are also continuous at the boundary. However,
curvature radius *R* is generally not continuous at the boundary
because of the change of the velocity gradient across the boundary.
Cervený (1981c) derives a general formula that relates
the curvature radii of wavefronts on the two sides of the boundary.
This general formula can be greatly simplified by considering
that the velocity field is continuous at the boundary
and that the boundary has a zero curvature. The result is

(11) |

vpbound
Ray-tracing across a boundary of two triangular cells. Because the velocity is
continuous at the boundary, the angle of incidence is equal to the
angle of refraction.
Figure 3 |

Finite difference methods propagate local wavefronts in cells of constant velocity. Because wavefronts propagate along straight rays in a constant velocity field, local ray-tracing is trivial. The local-plane-wave assumption implies that . All these simplifications make finite difference methods more efficient than the local ray-tracing method. Of course, they also reduce the accuracy of the computations.

11/17/1997